Group #23783358 — subgroups

Subgroups — C₄₁×#7157 #23783358

Order 10496.

214 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 214 conjugacy classes of subgroups — too many to draw. Only its 126 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄₁×#346	C₂	1	1	yes	maximal, characteristic
2	C₄₁×#1564	C₂	1	2	yes	maximal
3	C₂²×C₁₆×C₄₁	C₄	1	2	yes	—
4	C₄₁×#431	—	2	2	no	—
5	C₂×C₄×C₈×C₄₁	C₂²	1	1	yes	characteristic
6	C₄₁×#346	—	2	2	no	—
7	C₄₁×#335	—	2	2	no	—
8	C₄₁×#334	C₄	1	4	yes	—
9	C₂²×C₈×C₄₁	C₂×C₄	1	1	yes	characteristic
10	C₂²×C₈×C₄₁	Q₈	1	1	yes	characteristic
11	C₂×C₄²×C₄₁	D₄	1	1	yes	characteristic
12	C₂×C₁₆×C₄₁	—	2	2	no	—
13	C₂×C₁₆×C₄₁	—	2	2	no	—
14	C₂×C₁₆×C₄₁	C₈	1	4	yes	complemented
15	C₄⋊C₈×C₄₁	—	2	4	no	—
16	C₂²⋊C₈×C₄₁	—	4	2	no	—
17	C₄×C₈×C₄₁	—	4	2	no	—
18	C₄×C₈×C₄₁	C₂×C₄	1	2	yes	—
19	C₄×C₈×C₄₁	D₄	1	2	yes	—
20	C₂²×C₄×C₄₁	C₄⋊C₄	1	1	yes	characteristic
21	C₂²×C₄×C₄₁	D₈	1	1	yes	characteristic
22	C₂²×C₄×C₄₁	Q₁₆	1	1	yes	characteristic
23	C₂×C₈×C₄₁	—	2	2	no	—
24	C₂×C₈×C₄₁	—	4	4	no	—
25	C₂×C₈×C₄₁	C₄²	1	1	yes	characteristic
26	C₂×C₈×C₄₁	C₂²⋊C₄	1	1	yes	characteristic
27	C₂×C₈×C₄₁	C₄⋊C₄	1	2	yes	—
28	C₂×C₈×C₄₁	C₂×C₈	1	2	yes	—
29	C₂×C₈×C₄₁	M₁₆	1	2	yes	—
30	C₄²×C₄₁	C₂²⋊C₄	1	2	yes	—
31	C₄²×C₄₁	SD₁₆	1	2	yes	—
32	C₁₆×C₄₁	—	2	4	no	—
33	C₂³×C₄₁	#107	1	1	yes	characteristic
34	C₂×C₄×C₄₁	—	4	1	no	—
35	C₂×C₄×C₄₁	—	4	1	no	—
36	C₂×C₄×C₄₁	#95	1	1	yes	characteristic
37	C₂×C₄×C₄₁	#102	1	2	yes	—
38	C₂×C₄×C₄₁	#103	1	2	yes	—
39	C₂×C₄×C₄₁	C₄⋊C₈	1	2	yes	—
40	C₂×C₄×C₄₁	#106	1	1	yes	characteristic
41	C₂×C₄×C₄₁	#108	1	2	yes	—
42	C₈×C₄₁	—	2	2	no	—
43	C₈×C₄₁	—	8	4	no	—
44	C₈×C₄₁	C₄×C₈	1	1	yes	characteristic
45	C₈×C₄₁	#97	1	1	yes	characteristic
46	C₈×C₄₁	C₂²⋊C₈	1	2	yes	—
47	#7157	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
48	C₂²×C₄₁	—	2	1	no	—
49	C₂²×C₄₁	—	2	1	no	—
50	C₂²×C₄₁	#335	1	2	yes	—
51	C₂²×C₄₁	#340	1	1	yes	characteristic
52	C₄×C₄₁	—	2	2	no	—
53	C₄×C₄₁	—	2	2	no	—
54	C₄×C₄₁	#334	1	2	yes	—
55	C₄×C₄₁	#336	1	1	yes	characteristic
56	C₄×C₄₁	#341	1	1	yes	characteristic
57	C₂×#346	C₂×C₄₁	1	1	yes	characteristic
58	#1564	C₂×C₄₁	1	2	yes	—
59	C₂×C₄₁	—	2	2	no	—
60	C₂×C₄₁	#1277	1	1	yes	characteristic
61	C₂×C₄₁	#1384	1	1	yes	characteristic
62	C₂×C₄₁	#1385	1	1	yes	characteristic
63	C₂²×C₁₆	C₄×C₄₁	1	2	yes	—
64	#431	—	2	2	no	—
65	C₂×C₄×C₈	C₂²×C₄₁	1	1	yes	characteristic
66	#346	—	2	2	no	—
67	#335	—	2	2	no	—
68	#334	C₄×C₄₁	1	4	yes	—
69	C₄₁	#7157	1	1	yes	characteristic, direct product factor, Sylow
70	C₂²×C₈	C₂×C₄×C₄₁	1	1	yes	characteristic
71	C₂²×C₈	Q₈×C₄₁	1	1	yes	characteristic
72	C₂×C₄²	D₄×C₄₁	1	1	yes	characteristic
73	C₂×C₁₆	—	2	2	no	—
74	C₂×C₁₆	—	2	2	no	—
75	C₂×C₁₆	C₈×C₄₁	1	4	yes	complemented
76	C₄⋊C₈	—	2	4	no	—
77	C₂²⋊C₈	—	4	2	no	—
78	C₄×C₈	—	4	2	no	—
79	C₄×C₈	C₂×C₄×C₄₁	1	2	yes	—
80	C₄×C₈	D₄×C₄₁	1	2	yes	—
81	C₂²×C₄	C₄⋊C₄×C₄₁	1	1	yes	characteristic
82	C₂²×C₄	D₈×C₄₁	1	1	yes	characteristic
83	C₂²×C₄	Q₁₆×C₄₁	1	1	yes	characteristic
84	C₂×C₈	—	2	2	no	—
85	C₂×C₈	—	4	4	no	—
86	C₂×C₈	C₄²×C₄₁	1	1	yes	characteristic
87	C₂×C₈	C₂²⋊C₄×C₄₁	1	1	yes	characteristic
88	C₂×C₈	C₄⋊C₄×C₄₁	1	2	yes	—
89	C₂×C₈	C₂×C₈×C₄₁	1	2	yes	—
90	C₂×C₈	M₁₆×C₄₁	1	2	yes	—
91	C₄²	C₂²⋊C₄×C₄₁	1	2	yes	—
92	C₄²	SD₁₆×C₄₁	1	2	yes	—
93	C₁₆	—	2	4	no	—
94	C₂³	#107×C₄₁	1	1	yes	characteristic
95	C₂×C₄	—	4	1	no	—
96	C₂×C₄	—	4	1	no	—
97	C₂×C₄	#95×C₄₁	1	1	yes	characteristic
98	C₂×C₄	#102×C₄₁	1	2	yes	—
99	C₂×C₄	#103×C₄₁	1	2	yes	—
100	C₂×C₄	C₄⋊C₈×C₄₁	1	2	yes	—
101	C₂×C₄	#106×C₄₁	1	1	yes	characteristic
102	C₂×C₄	#108×C₄₁	1	2	yes	—
103	C₈	—	2	2	no	—
104	C₈	—	8	4	no	—
105	C₈	C₄×C₈×C₄₁	1	1	yes	characteristic
106	C₈	#97×C₄₁	1	1	yes	characteristic
107	C₈	C₂²⋊C₈×C₄₁	1	2	yes	—
108	C₂²	—	2	1	no	—
109	C₂²	—	2	1	no	—
110	C₂²	C₄₁×#335	1	2	yes	—
111	C₂²	C₄₁×#340	1	1	yes	characteristic
112	C₄	—	2	2	no	—
113	C₄	—	2	2	no	—
114	C₄	C₄₁×#334	1	2	yes	—
115	C₄	C₄₁×#336	1	1	yes	characteristic
116	C₄	C₄₁×#341	1	1	yes	characteristic
117	C₂	—	2	2	no	—
118	C₂	C₄₁×#1277	1	1	yes	characteristic
119	C₂	C₄₁×#1384	1	1	yes	characteristic
120	C₂	C₄₁×#1385	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄₁×#346

C₂

yes

maximal, characteristic

C₄₁×#1564

C₂

yes

maximal

C₂²×C₁₆×C₄₁

C₄

yes

—

C₄₁×#431

—

C₂×C₄×C₈×C₄₁

C₂²

yes

characteristic

C₄₁×#346

—

C₄₁×#335

—

C₄₁×#334

C₄

yes

—

C₂²×C₈×C₄₁

C₂×C₄

yes

characteristic

C₂²×C₈×C₄₁

Q₈

yes

characteristic

C₂×C₄²×C₄₁

D₄

yes

characteristic

C₂×C₁₆×C₄₁

—

C₂×C₁₆×C₄₁

—

C₂×C₁₆×C₄₁

C₈

yes

complemented

C₄⋊C₈×C₄₁

—

C₂²⋊C₈×C₄₁

—

C₄×C₈×C₄₁

—

C₄×C₈×C₄₁

C₂×C₄

yes

—

C₄×C₈×C₄₁

D₄

yes

—

C₂²×C₄×C₄₁

C₄⋊C₄

yes

characteristic

C₂²×C₄×C₄₁

D₈

yes

characteristic

C₂²×C₄×C₄₁

Q₁₆

yes

characteristic

C₂×C₈×C₄₁

—

C₂×C₈×C₄₁

—

C₂×C₈×C₄₁

C₄²

yes

characteristic

C₂×C₈×C₄₁

C₂²⋊C₄

yes

characteristic

C₂×C₈×C₄₁

C₄⋊C₄

yes

—

C₂×C₈×C₄₁

C₂×C₈

yes

—

C₂×C₈×C₄₁

M₁₆

yes

—

C₄²×C₄₁

C₂²⋊C₄

yes

—

C₄²×C₄₁

SD₁₆

yes

—

C₁₆×C₄₁

—

C₂³×C₄₁

#107

yes

characteristic

C₂×C₄×C₄₁

—

C₂×C₄×C₄₁

—

C₂×C₄×C₄₁

#95

yes

characteristic

C₂×C₄×C₄₁

#102

yes

—

C₂×C₄×C₄₁

#103

yes

—

C₂×C₄×C₄₁

C₄⋊C₈

yes

—

C₂×C₄×C₄₁

#106

yes

characteristic

C₂×C₄×C₄₁

#108

yes

—

C₈×C₄₁

—

C₈×C₄₁

—

C₈×C₄₁

C₄×C₈

yes

characteristic

C₈×C₄₁

#97

yes

characteristic

C₈×C₄₁

C₂²⋊C₈

yes

—

#7157

C₄₁

yes

maximal, characteristic, direct product factor, Sylow

C₂²×C₄₁

—

C₂²×C₄₁

—

C₂²×C₄₁

#335

yes

—

C₂²×C₄₁

#340

yes

characteristic

C₄×C₄₁

—

C₄×C₄₁

—

C₄×C₄₁

#334

yes

—

C₄×C₄₁

yes

C₄×C₄₁

yes

C₂×C₄₁

yes

characteristic

#1564

C₂×C₄₁

yes

—

C₂×C₄₁

—

C₂×C₄₁

yes

C₂×C₄₁

yes

C₂×C₄₁

yes

C₂²×C₁₆

C₄×C₄₁

yes

—

#431

—

C₂×C₄×C₈

C₂²×C₄₁

yes

—

—

C₄×C₄₁

yes

—

C₄₁

#7157

yes

characteristic, direct product factor, Sylow

C₂²×C₈

C₂×C₄×C₄₁

yes

characteristic

C₂²×C₈

Q₈×C₄₁

yes

characteristic

C₂×C₄²

D₄×C₄₁

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₈×C₄₁

yes

complemented

C₄⋊C₈

—

C₂²⋊C₈

—

C₄×C₈

—

C₄×C₈

C₂×C₄×C₄₁

yes

—

C₄×C₈

D₄×C₄₁

yes

—

C₂²×C₄

C₄⋊C₄×C₄₁

yes

characteristic

C₂²×C₄

D₈×C₄₁

yes

characteristic

C₂²×C₄

Q₁₆×C₄₁

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₄²×C₄₁

yes

characteristic

C₂×C₈

C₂²⋊C₄×C₄₁

yes

characteristic

C₂×C₈

C₄⋊C₄×C₄₁

yes

—

C₂×C₈

C₂×C₈×C₄₁

yes

—

C₂×C₈

M₁₆×C₄₁

yes

—

C₄²

C₂²⋊C₄×C₄₁

yes

—

C₄²

SD₁₆×C₄₁

yes

—

C₁₆

—

C₂³

#107×C₄₁

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

#95×C₄₁

yes

characteristic

C₂×C₄

#102×C₄₁

yes

—

C₂×C₄

#103×C₄₁

yes

—

100

C₂×C₄

C₄⋊C₈×C₄₁

yes

—

101

C₂×C₄

#106×C₄₁

yes

characteristic

102

C₂×C₄

#108×C₄₁

yes

—

103

C₈

—

104

C₈

—

105

C₈

C₄×C₈×C₄₁

yes

characteristic

106

C₈

#97×C₄₁

yes

characteristic

107

C₈

C₂²⋊C₈×C₄₁

yes

—

108

C₂²

—

109

C₂²

—

110

C₂²

C₄₁×#335

yes

—

111

C₂²

C₄₁×#340

yes

characteristic

112

C₄

—

113

C₄

—

114

C₄

C₄₁×#334

yes

—

115

C₄

C₄₁×#336

yes

characteristic

116

C₄

C₄₁×#341

yes

characteristic

117

C₂

—

118

C₂

C₄₁×#1277

yes

characteristic

119

C₂

C₄₁×#1384

yes

characteristic

120

C₂

C₄₁×#1385

yes

characteristic

	subgroup	G/H
center	C₂×C₄×C₄₁	#106
commutator	C₈	C₄×C₈×C₄₁
Frattini, cosocle	C₂×C₄×C₈	C₂²×C₄₁
socle	C₂²×C₄₁	#340
prime radical	C₂³×C₄₁	#107
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₄₁ ◁ C₂×C₄²×C₄₁ ◁ C₂×C₄×C₈×C₄₁ ◁ G

subgroup

G/H

center

C₂×C₄×C₄₁

#106

commutator

C₈

C₄×C₈×C₄₁

Frattini, cosocle

C₂×C₄×C₈

C₂²×C₄₁

C₂²×C₄₁

C₂³×C₄₁

G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂×C₄×C₄₁ ◁ C₂×C₄²×C₄₁ ◁ C₂×C₄×C₈×C₄₁ ◁ G

N	H ≅ G/N	type
14. C₂×C₁₆×C₄₁	C₈	semidirect
47. #7157	C₄₁	direct (N × H)
69. C₄₁	#7157	direct (N × H)
75. C₂×C₁₆	C₈×C₄₁	semidirect

H ≅ G/N

type

14. C₂×C₁₆×C₄₁

C₈

semidirect

47. #7157

C₄₁

direct (N × H)

69. C₄₁

#7157

direct (N × H)

75. C₂×C₁₆

C₈×C₄₁

semidirect

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×C₄₁×#346	2
C₄₁×#1564	2
#7157	41

subgroup

index

C₂×C₄₁×#346

C₄₁×#1564

#7157

Subgroups — C₄₁×#7157 #23783358

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C41×#7157 #23783358

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×#7157 #23783358